Solve the Inequality for y y/4+2-(y+1)/6>=(y+5)/9

$\frac{y}{4} + 2 - \frac{y + 1}{6} \geq \frac{y + 5}{9}$

Step 1

Multiply both sides by $9$ .

$(\frac{y}{4} + 2 - \frac{y + 1}{6}) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Step 2

Simplify.

Simplify the left side.

Simplify $(\frac{y}{4} + 2 - \frac{y + 1}{6}) \cdot 9$ .

To write $\frac{y}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{y}{4} \cdot \frac{3}{3} - \frac{y + 1}{6} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

To write $- \frac{y + 1}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(\frac{y}{4} \cdot \frac{3}{3} - \frac{y + 1}{6} \cdot \frac{2}{2} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{y}{4}$ by $\frac{3}{3}$ .

$(\frac{y \cdot 3}{4 \cdot 3} - \frac{y + 1}{6} \cdot \frac{2}{2} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Multiply $4$ by $3$ .

$(\frac{y \cdot 3}{12} - \frac{y + 1}{6} \cdot \frac{2}{2} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Multiply $\frac{y + 1}{6}$ by $\frac{2}{2}$ .

$(\frac{y \cdot 3}{12} - \frac{(y + 1) \cdot 2}{6 \cdot 2} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Multiply $6$ by $2$ .

$(\frac{y \cdot 3}{12} - \frac{(y + 1) \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Combine the numerators over the common denominator.

$(\frac{y \cdot 3 - (y + 1) \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Simplify the numerator.

Move $3$ to the left of $y$ .

$(\frac{3 \cdot y - (y + 1) \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Apply the distributive property.

$(\frac{3 y + (- y - 1 \cdot 1) \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Multiply $- 1$ by $1$ .

$(\frac{3 y + (- y - 1) \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Apply the distributive property.

$(\frac{3 y - y \cdot 2 - 1 \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Multiply $2$ by $- 1$ .

$(\frac{3 y - 2 y - 1 \cdot 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Multiply $- 1$ by $2$ .

$(\frac{3 y - 2 y - 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Subtract $2 y$ from $3 y$ .

$(\frac{y - 2}{12} + 2) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$(\frac{y - 2}{12} + 2 \cdot \frac{12}{12}) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Simplify terms.

Combine $2$ and $\frac{12}{12}$ .

$(\frac{y - 2}{12} + \frac{2 \cdot 12}{12}) \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Combine the numerators over the common denominator.

$\frac{y - 2 + 2 \cdot 12}{12} \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Simplify the numerator.

Multiply $2$ by $12$ .

$\frac{y - 2 + 24}{12} \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Add $- 2$ and $24$ .

$\frac{y + 22}{12} \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Simplify terms.

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$\frac{y + 22}{3 (4)} \cdot 9 \geq \frac{y + 5}{9} \cdot 9$

Factor $3$ out of $9$ .

$\frac{y + 22}{3 \cdot 4} \cdot (3 \cdot 3) \geq \frac{y + 5}{9} \cdot 9$

Cancel the common factor.

$\frac{y + 22}{3 \cdot 4} \cdot (3 \cdot 3) \geq \frac{y + 5}{9} \cdot 9$

Rewrite the expression.

$\frac{y + 22}{4} \cdot 3 \geq \frac{y + 5}{9} \cdot 9$

Combine $\frac{y + 22}{4}$ and $3$ .

$\frac{(y + 22) \cdot 3}{4} \geq \frac{y + 5}{9} \cdot 9$

Move $3$ to the left of $y + 22$ .

$\frac{3 (y + 22)}{4} \geq \frac{y + 5}{9} \cdot 9$

Simplify the right side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{3 (y + 22)}{4} \geq \frac{y + 5}{9} \cdot 9$

Rewrite the expression.

$\frac{3 (y + 22)}{4} \geq y + 5$

Step 3

Solve for $y$ .

Multiply both sides by $4$ .

$\frac{3 (y + 22)}{4} \cdot 4 \geq (y + 5) \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{3 (y + 22)}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 (y + 22)}{4} \cdot 4 \geq (y + 5) \cdot 4$

Rewrite the expression.

$3 (y + 22) \geq (y + 5) \cdot 4$

Apply the distributive property.

$3 y + 3 \cdot 22 \geq (y + 5) \cdot 4$

Multiply $3$ by $22$ .

$3 y + 66 \geq (y + 5) \cdot 4$

Simplify the right side.

Simplify $(y + 5) \cdot 4$ .

Apply the distributive property.

$3 y + 66 \geq y \cdot 4 + 5 \cdot 4$

Simplify the expression.

Move $4$ to the left of $y$ .

$3 y + 66 \geq 4 \cdot y + 5 \cdot 4$

Multiply $5$ by $4$ .

$3 y + 66 \geq 4 y + 20$

Solve for $y$ .

Move all terms containing $y$ to the left side of the inequality.

Subtract $4 y$ from both sides of the inequality.

$3 y + 66 - 4 y \geq 20$

Subtract $4 y$ from $3 y$ .

$- y + 66 \geq 20$

Move all terms not containing $y$ to the right side of the inequality.

Subtract $66$ from both sides of the inequality.

$- y \geq 20 - 66$

Subtract $66$ from $20$ .

$- y \geq - 46$

Divide each term in $- y \geq - 46$ by $- 1$ and simplify.

Divide each term in $- y \geq - 46$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \leq \frac{- 46}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} \leq \frac{- 46}{- 1}$

Divide $y$ by $1$ .

$y \leq \frac{- 46}{- 1}$

Simplify the right side.

Divide $- 46$ by $- 1$ .

$y \leq 46$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$y \leq 46$

Interval Notation:

$(- \infty, 46]$

Solve the Inequality for y y/4+2-(y+1)/6>=(y+5)/9

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